Page No 14:
Write the following in decimal form and say what kind of decimal expansion each has:
(i) (ii) (iii)
(iv) (v) (vi)
You know that. Can you predict what the decimal expansion of are, without actually doing the long division? If so, how?
[Hint: Study the remainders while finding the value of carefully.]
Yes. It can be done as follows.
Express the following in the form, where p and q are integers and q ≠ 0.
(i) (ii) (iii)
Let x = 0.666…
10x = 6.666…
10x = 6 + x
9x = 6
Let x = 0.777…
10x = 7.777…
10x = 7 + x
Let x = 0.001001…
1000x = 1.001001…
1000x = 1 + x
999x = 1
Express 0.99999…in the form. Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.
Let x = 0.9999…
10x = 9.9999…
10x = 9 + x
9x = 9
x = 1
What can the maximum number of digits be in the repeating block of digits in the decimal expansion of? Perform the division to check your answer.
It can be observed that,
There are 16 digits in the repeating block of the decimal expansion of
Look at several examples of rational numbers in the form (q ≠ 0), where p and q are integers with no common factors other than 1 and having terminating decimal representations (expansions). Can you guess what property q must satisfy?
Terminating decimal expansion will occur when denominator q of rational number is either of 2, 4, 5, 8, 10, and so on…
It can be observed that terminating decimal may be obtained in the situation where prime factorisation of the denominator of the given fractions has the power of 2 only or 5 only or both.
Write three numbers whose decimal expansions are non-terminating non-recurring.
3 numbers whose decimal expansions are non-terminating non-recurring are as follows.
Find three different irrational numbers between the rational numbers and.
3 irrational numbers are as follows.
Classify the following numbers as rational or irrational:
(i) (ii) (iii) 0.3796
(iv) 7.478478 (v) 1.101001000100001…
As the decimal expansion of this number is non-terminating non-recurring, therefore, it is an irrational number.
It is a rational number as it can be represented in form.
As the decimal expansion of this number is terminating, therefore, it is a rational number.
(iv) 7.478478 …
As the decimal expansion of this number is non-terminating recurring, therefore, it is a rational number.
(v) 1.10100100010000 …
As the decimal expansion of this number is non-terminating non-repeating, therefore, it is an irrational number.